Rationalize

Finds a simple rational number near x.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageRationalize✓↗

Rationalize(x)the simplest rational within about one machine epsilon of x.

Rationalize(x, tolerance)the simplest rational within the given tolerance of x.

Rationalize(x, tolerance?)Finds a simple rational number near x.

Typed signatures

Type	Package	Applies
(real, real<0..>?) -> rational	compute-engine	always
(value, number?) -> value	analyticreplaces compute-engine	always
(value, number?) -> value	combinatoricsreplaces analytic	always

Domain: Arithmetic

Details

• Finds a simple rational number near x.
• With one argument, it finds the simplest fraction within about one machine epsilon of x's current value; a second argument sets an explicit tolerance instead.
• An already-exact rational (or integer) is returned unchanged.
• Builds the approximation via continued fractions, the same technique behind classic approximations like $\pi \approx 355/113$.

Examples

3 more kept as data, checked against the oracles

Basic #

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In$\mathrm{Rationalize}(0.1)$Out$\frac{1}{10}$

An exact integer is returned unchanged

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In$\mathrm{Rationalize}(2)$Out$2$

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In$\mathrm{Rationalize}(2.5)$Out$\frac{5}{2}$

Recognizes a float that is, to machine precision, a simple fraction

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In$\mathrm{Rationalize}(0.333\,333\,333\,333\,333\,3)$Out$\frac{1}{3}$

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In$\mathrm{Rationalize}(6.75)$Out$\frac{27}{4}$

A second argument sets the tolerance

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In$\mathrm{Rationalize}(π, 0.01)$Out$\frac{22}{7}$

Scope #

Any exact numeric expression, to any tolerance

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In$\mathrm{Rationalize}(e^{\sqrt{2}}, 2^{-12})$Out$\frac{218}{53}$

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In$\mathrm{Rationalize}(\sqrt{2}, 0.001)$Out$\frac{41}{29}$

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In$\mathrm{Rationalize}(-0.125)$Out$-\frac{1}{8}$

The rational with the smallest denominator within 0.001 of $\pi$ -- $201/64$, not compute-engine's own closer-but-larger-denominator continued-fraction convergent $333/106$

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In$\mathrm{Rationalize}(π, 0.001)$Out$\frac{201}{64}$

A tolerance finer than a double can resolve is still honoured -- the simplest fraction within $10^{-16}$ of $\pi$ itself, not of $N(\pi)$, which is already $1.2 \times 10^{-16}$ away

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In$\mathrm{Rationalize}(π, 1.0\cdot10^{-16})$Out$\frac{245\,850\,922}{78\,256\,779}$

Rationalizes every approximate number inside an expression

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In$\mathrm{Rationalize}(1.2+6.7x)$Out$\frac{67x}{10}+\frac{6}{5}$

Threads element-wise over a list; compute-engine's own Rationalize takes only a single real

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In$\mathrm{Rationalize}(\bigl\lbrack0.5, 0.25, 0.2\bigr\rbrack)$Out$\bigl\lbrack\frac{1}{2}, \frac{1}{4}, \frac{1}{5}\bigr\rbrack$

A zero tolerance gives the exact rational the machine double denotes -- $N(\pi)$'s own dyadic value, $884279719003555/2^{48}$, not merely a rational within some epsilon of it

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In$\mathrm{Rationalize}(N(π), 0)$Out$\frac{884\,279\,719\,003\,555}{281\,474\,976\,710\,656}$

Applications #

Recovers a simple fraction from a decimal measurement

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In$\mathrm{Rationalize}(0.75)$Out$\frac{3}{4}$

Recognizes a ratio of consecutive Fibonacci numbers in a decimal approximation of $1/\varphi$

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In$\mathrm{Rationalize}(0.618\,034, 0.000\,1)$Out$\frac{55}{89}$

Properties #

With an explicit tolerance, recovers a classic approximation: 22/7 is within 0.01 of π

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In$\mathrm{Rationalize}(π, 0.01)=\frac{22}{7}$Out$\mathrm{True}$

Possible issues #

Without enough repeating digits to reach machine precision, it just returns the exact decimal as a fraction rather than snapping to 1/3

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In$\mathrm{Rationalize}(0.333\,333)$Out$\frac{333\,333}{1\,000\,000}$

A loose tolerance can snap to something far simpler -- 0 is within 0.5 of 0.1

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In$\mathrm{Rationalize}(0.1, 0.5)$Out$0$

Without a tolerance, a float with no simple fraction nearby still becomes a fraction with a large denominator

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In$\mathrm{Rationalize}(N(π))$Out$\frac{80\,143\,857}{25\,510\,582}$

Neat examples #

Milü, Zu Chongzhi's 5th-century approximation of π, accurate to six decimal places

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In$\mathrm{Rationalize}(π, 0.000\,001)$Out$\frac{355}{113}$

About

Examples	24
Checked against	Wolfram Language 24 agree
Crosswalk	Wolfram Language

See also: Round